Alexandroff locales

Definition

objects are Alexandroff locales, that is locales of the form AlexPAlex P for P∈PosetP\in Poset with Open(Alex(P))=UpSets(P)Open(Alex(P)) = UpSets(P);

morphisms are those morphisms of locales f:AlexP→AlexQf\colon Alex P \to Alex Q, for which the dual inverse image morphism of framesf*:UpSet(Q)→UpSet(P)f^*\colon UpSet(Q) \to UpSet(P) has a left adjointf!:UpSet(P)→UpSet(Q)f_!\colon UpSet(P) \to UpSet(Q).